Internal rate of return

The internal rate of return (IRR) or economic rate of
return (ERR) is a rate of return used in capital budgeting
to measure and compare the proﬁtability of investments.
It is also called the discounted cash ﬂow rate of return
(DCFROR).[1] In the context of savings and loans the
IRR is also called the eﬀective interest rate. The term
internal refers to the fact that its calculation does not incorporate environmental factors (e.g., the interest rate or
inﬂation).

1

rate of return is greater than an established minimum
acceptable rate of return or cost of capital. In a scenario where an investment is considered by a ﬁrm that
has shareholders, this minimum rate is the cost of capital
of the investment (which may be determined by the riskadjusted cost of capital of alternative investments). This
ensures that the investment is supported by equity holders since, in general, an investment whose IRR exceeds
its cost of capital adds value for the company (i.e., it is
economically proﬁtable).
One of the uses of IRR is by corporations that wish to
compare capital projects. For example, a corporation will
evaluate an investment in a new plant versus an extension
of an existing plant based on the IRR of each project.
In such a case, each new capital project must produce
an IRR that is higher than the company’s cost of capital.
Once this hurdle is surpassed, the project with the highest
IRR would be the wiser investment, all other things being
equal (including risk).

Deﬁnition

The internal rate of return on an investment or project
is the “annualized eﬀective compounded return rate” or
rate of return that makes the net present value (NPV as
NET*1/(1+IRR)^year) of all cash ﬂows (both positive
and negative) from a particular investment equal to zero.
It can also be deﬁned as the discount rate at which the
present value of all future cash ﬂow is equal to the initial IRR is also useful for corporations in evaluating stock
investment or in other words the rate at which an invest- buyback programs. Clearly, if a company allocates a
ment breaks even.
substantial amount to a stock buyback, the analysis must
In more speciﬁc terms, the IRR of an investment is the show that the company’s own stock is a better investment
discount rate at which the net present value of costs (neg- (has a higher IRR) than any other use of the funds for
ative cash ﬂows) of the investment equals the net present other capital projects, or than any acquisition candidate
value of the beneﬁts (positive cash ﬂows) of the invest- at current market prices.
ment.
IRR calculations are commonly used to evaluate the
desirability of investments or projects. The higher a 3 Calculation
project’s IRR, the more desirable it is to undertake the
project. Assuming all projects require the same amount Given a collection of pairs (time, cash ﬂow) involved in
of up-front investment, the project with the highest IRR a project, the internal rate of return follows from the net
would be considered the best and undertaken ﬁrst.
present value as a function of the rate of return. A rate of
A ﬁrm (or individual) should, in theory, undertake all return for which this function is zero is an internal rate of
projects or investments available with IRRs that exceed return.
the cost of capital. Investment may be limited by avail- Given the (period, cash ﬂow) pairs ( n , Cn ) where n is a
ability of funds to the ﬁrm and/or by the ﬁrm’s capacity positive integer, the total number of periods N , and the
or ability to manage numerous projects.
net present value NPV , the internal rate of return is given
by r in:

2

Uses of IRR
NPV =

Because the internal rate of return is a rate quantity, it is
an indicator of the eﬃciency, quality, or yield of an investment. This is in contrast with the net present value,
which is an indicator of the value or magnitude of an investment.

N
∑

Cn
=0
(1
+
r)n
n=0

The period is usually given in years, but the calculation
may be made simpler if r is calculated using the period
in which the majority of the problem is deﬁned (e.g., using months if most of the cash ﬂows occur at monthly
An investment is considered acceptable if its internal intervals) and converted to a yearly period thereafter.
1

followed by multiple
as in the case of a life annuity.
This r can be found to an arbitrary degree of accuracy.
into the above formula.
Example
If an investment may be given by the sequence of cash Given two estimates r1 and r2 for IRR.
(which is most accurate when 0 > NPVn > NPVn−1
) has been shown to be almost 10 times more accurate
than the secant formula for a wide range of interest rates
3.
Of particular interest is the case where the stream of payIn the case that the cash ﬂows are random variables. 3. using the stream of
Since the above is a manifestation of the general problem payments {−4000.
then the sequence converges to one of the roots. set C0 = 0 and compute NPV).1 Numerical solution
and initial guesses. the value of r cannot be found analytically.
3.2 the secant formula with
are many numerical methods that can be used to estimate correction gives an IRR estimate of 14. the answer is 5.
An accuracy of 0. 1410. 1200... More accurate
interpolation formulas can also be obtained: for instance
the secant formula with correction
36200
54800
48100
NPV = −123400+
+
+
= 0. there guesses r1 = 0. This is sometimes referred to as the
then the IRR r is given by
Hit and Trial (or Trial and Error) method.2% (0. the secant method
equation (see above) with n = 2 always produces an imﬂows
proved estimate r3 .
.00001% is provided by Microsoft Excel..in )
Having r1 >r0 when NPV0 > 0 or r1 <r0 when NPV0 < 0 Here..
• If function NPV(i) has no real roots.rn .2
3 CALCULATION
Any ﬁxed time can be used in place of the present (e.2 Numerical solution for single outﬂow and
the end of one interval of an annuity).
may speed up convergence of rn to r .. the value obtained
multiple inﬂows
is zero if and only if the NPV is zero. numerical methods or graphical methods must be
used. NPV1.
strictly decreasing function of interest rate.1.
r2 = (1 + r1 )p − 1
where
A = inﬂows of sum = C1 + · · · + CN
p=
log(A/|C0 |)
.
If applied iteratively. then the sequence tends towards +∞. the expected values are put
inﬂows occurring at equal periods.in refers to the NPV of the inﬂows only (that
is.g. and
changing the values of the initial pairs may change
the root to which it converges.
this corresponds to:
Often. such
ments consists of a single outﬂow.r2 .
C0 < 0.
log(A/NPV1.
(1 + r)1 (1 + r)2 (1 + r)3
(
)(
rn − rn−1
NPVn−1
r
=
r
−NPV
1 − 1. using the secant method. r is given by
method.4
n
n
In this case. r = .7% error)
as compared to IRR = 13.0596).25 and r2 = 0. The following initial guesses may
be used:
r1 = (A/|C0 |)
2/(N +1)
−1
The convergence behaviour of by the following:
• If the function NPV(i) has a single real root r . then
the sequence converges reproducibly towards r . 1050} and initial
of ﬁnding the roots of the equation NPV(r) = 0 . 1875. Cn ≥ 0 for n ≥ 1. There is always a single unique solution for IRR. For example. that n+1
NPVn − NPVn−1
NPVn−1 − 3NPVn +
is.96% (in the calculation. Other improved formulas may be found in [2]
(
rn+1 = rn − NPVn
rn − rn−1
NPVn − NPVn−1
)
.2% (7% error) from the secant
r . either the secant method or the improved formula always converges to the correct solution.
• If the function NPV(i) has n real roots r1 .
where rn is considered the n th approximation of the IRR. In the above notation.1
In this case the NPV of the payment stream is a convex.1. For example.
Both the secant method and the improved formula rely on
initial guesses for IRR. In this
case.

rate of return
In general. if the ﬁrst and last cash
Mathematically. the value of the investment is assumed to
undergo exponential growth or decay according to some
rate of return (any value greater than −100%).
even though its IRR (= x-axis intercept) is lower than for project
'B' (click to enlarge)
In cases where one project has a higher initial investment
than a second mutually exclusive project. as a measure of investment eﬃciency may give better insights in
capital constrained situations. However. but
If the IRR is greater than the cost of capital. This applies for
example when a customer makes a deposit before a speciﬁc machine is built.
IRR should not be used to compare projects of diﬀerent duration. and the IRR of a series of cash
Modiﬁed Internal Rate of Return (MIRR) considers cost ﬂows is deﬁned as any rate of return that results in a net
of capital. so a high rate of return is best. and towards a rate of return of positive inﬁnity
the net present value approaches the ﬁrst cash ﬂow (the
one at the present).
Thus.2
Decision criterion
In a series of cash ﬂows like (−10.
See also [5] for a way of identifying the relevant value of
the IRR from a set of multiple IRR solutions. a rate of return that
of a project’s probable return. accept the then receives more than one possesses. MIRR is used. so now a low rate of return is best.
When a project has multiple IRRs it may be more convenient to compute the IRR of the project with the beneﬁts
reinvested. Therefore. In general the
not be used to rate mutually exclusive projects. with discontinuities for cash ﬂows. the net present value added
by a project with longer duration but lower IRR could be
greater than that of a project of similar size. However. usually equal to the project’s
cost of capital. In this case
If the IRR is less than the cost of capital.[6] Apparently.
where there is usually a large cash outﬂow at the end of
4 Problems with using internal the project.
Despite a strong academic preference for NPV. but with shorter duration and higher
IRR. one initially invests money. Sturm’s theorem can be used to determine
As an investment decision tool. Project 'A' has a higher NPV (for certain discount rates). but only IRR equation cannot be solved analytically but only iteratively.
5 Mathematics
In the case of positive cash ﬂows followed by negative
ones and then by positive ones (for example.
like in the example 0% as well as 10%. For example. −11).
money. so then one owes
project.
results in the correct value of zero after the last cash ﬂow). the IRR can be calculated by solving a polynomial equation. the calculated IRR should if that equation has a unique real solution.3
3. when comparing mutually exclusive projects. NPV is the appropriate
measure. surveys
indicate that executives prefer IRR over NPV. NPV remains the “more accurate” reﬂection of value to the business. the IRR approach can still be interpreted in a way
that is consistent with the present value approach provided that the underlying investment stream is correctly
identiﬁed as net investment or net borrowing. Towards a rate of return of −100% the net
present value approaches inﬁnity with the sign of the last
cash ﬂow. managers ﬁnd it easier to compare investments of
diﬀerent sizes in terms of percentage rates of return than
by dollars of NPV. internal rate(s) of return follow from the net present
value as a function of the rate of return. but a higher
NPV (increase in shareholders’ wealth) and should thus
be accepted over the second project (assuming no capital
constraints). which has an
assumed reinvestment rate.
NPV vs discount rate comparison for two mutually exclusive
projects. + + − − − +)
the IRR may have multiple values. the ﬁrst project
may have a lower IRR (expected return). In this case a discount
rate may be used for the borrowing cash ﬂow and the IRR
calculated for the investment cash ﬂow. 21. in terms of
total net cash ﬂows.
to decide whether a single project is worth investing in.[3] Accordingly. reject the it is not even clear whether a high or a low IRR is better. There may even be multiple IRRs for a single project.
.
project. This function is
continuous. Examples of this
type of project are strip mines and nuclear power plants. IRR.
It has been shown[4] that with multiple internal rates of
return. and is intended to provide a better indication present value of zero (or equivalently.

• Capital budgeting
This misconception likely stems from the modiﬁed internal rate of return MIRR concept. There is only
• Accounting rate of return
one unknown variable in the equation. such as the internal rate of return
as deﬁned above.
ual investor’s 401(k) or brokerage account. “The IRR is the annual interest rate of the ﬁxed
• (−1. a quadratic function of the discount
ance as the actual investment. when subjected to the same deposits and withfunction of 1/(1 + r). If the reinvestment rate is set at IRR. The term
internal rate of return or IRR or Since Inception Internal
6 The reinvestment misconception Rate of Return (SI-IRR) is in some contexts used to refer
to the unannualized return over the period. the IRR
In the case of a series of exclusively negative cash ﬂows calculation assumes that the same interest rate that is paid
followed by a series of exclusively positive ones. This
• Modiﬁed internal rate of return
is hardly a surprise .ple solutions issue disappears.
cash ﬂow (when the rate of return approaches inﬁnity).
8 Unannualized internal rate of return
Finally. the NPV is a quadratic
which.[10][11] The resulting rate is
function itself is not necessarily monotonically decreasing called the ﬁxed rate equivalent (FREQ). has the same ending balput diﬀerently. also many sources dis• Simple Dietz method
puting the so-called reinvestment assumption. 1.
Similarly. the re.[9] In those cases.[8]
. This is a misconception. Although the NPV. however.” This ﬁxed rate account is
rate r/(1 + r). by Descartes’ rule of signs.
changes in sign of cash ﬂow. IRR is simply the 9 See also
solution to the equation in the example shown above. the MIRR equals the IRR. subsequent investment.compounding cash ﬂows (with the
• Modiﬁed Dietz method
IRR) and then discounting them using the same discount
factor (the IRR) is obviously a zero-sum game.on positive balances is charged on negative balances. the number of internal rates of return can never be more than the number of
In the context of investment performance measurement. For this scenario. as is the case in real life. for r =
also called the replicating ﬁxed rate account for the invest100%. particularly
for periods of less than a year. It
sulting function of the rate of return is continuous and has been shown that this way of charging interest is the
monotonically decreasing from positive inﬁnity (when the root cause of the IRR’s multiple solutions problem. the multithe IRR is also unique (and equal). an equivalent. the highest NPV is −0. The NPV is set at zero.75. which allows for inclu• Discounted cash ﬂow
sion of a second. Examples of time series without an IRR:
9 SEE ALSO
7 The internal rate of return in
personal ﬁnance
The IRR can be used to measure the money-weighted
• Only negative cash ﬂows — the NPV is negative for
performance of ﬁnancial investments such as an individevery rate of return.
• Net present value
There are many. namely r. ing over time) is charged on negative balances.[9]
on its whole domain. and a holding period return. rather small positive cash ﬂow between
rate account (like a somewhat idealized savings account)
two negative cash ﬂows. Hence.[12]
It is often stated that IRR assumes reinvestment of all cash
ﬂows until the very end of the project.[9] more intuitive deﬁnition of the
IRR is.
ment. so an externally supplied cost of borrowing (possibly varythere is a unique rate of return for which it is zero. There are. The
cash ﬂows are static. −1).[10][11]
rate of return approaches −100%) to the value of the ﬁrst If the model is modiﬁed so that. in the case of a series of exclusively positive
cash ﬂows followed by a series of exclusively negative
ones the IRR is also unique. or
drawals as the actual investment. where r is the rate of return. highly reputable sources [3][7] arguing
that there is a hidden reinvestment assumption in the IRR
• Rate of return
calculation.
there is sometimes ambiguity in terminology between the
periodic rate of return. There is no hidden reinvestment assumption
associated with the calculation of IRR. it is at the IRR.4
ﬂow have a diﬀerent sign there exists an internal rate of
return. There are examples where the replicating ﬁxed rate
account encounters negative balances despite the fact that
the actual investment did not.